%I #31 May 28 2022 08:06:47
%S 1,5,33,85,161,261,385,533,705,901,1121,1365,1633,1925,2241,2581,2945,
%T 3333,3745,4181,4641,5125,5633,6165,6721,7301,7905,8533,9185,9861,
%U 10561,11285,12033,12805,13601,14421,15265,16133,17025,17941,18881,19845,20833
%N a(n) = 12*n^2 - 8*n + 1.
%C Sequence found by reading the line from 1, in the direction 1, 5, and the same line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, May 08 2018
%H Harvey P. Dale, <a href="/A185212/b185212.txt">Table of n, a(n) for n = 0..1000</a>
%H Leo Tavares, <a href="/A185212/a185212.jpg">Illustration: Square Block Rays</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*A000567(n) + 1.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=1, a(1)=5, a(2)=33. - _Harvey P. Dale_, Jul 07 2015
%F G.f.: (-1 - 2*x - 21*x^2)/(-1+x)^3. - _Harvey P. Dale_, Jul 07 2015
%F E.g.f.: (12*x^2 + 4*x + 1)*exp(x). - _G. C. Greubel_, Jun 25 2017
%F a(n) = A016754(n-1) + 4*A000384(n). - _Leo Tavares_, May 21 2022
%F From _Amiram Eldar_, May 28 2022: (Start)
%F Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 - 3*log(3)/8 + 1.
%F Sum_{n>=0} (-1)^n/a(n) = Pi/8 - sqrt(3)*arccoth(sqrt(3))/2 + 1. (End)
%t Table[12n^2-8n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,5,33},50] (* _Harvey P. Dale_, Jul 07 2015 *)
%o (Haskell)
%o a185212 = (+ 1) . (* 4) . a000567
%o (PARI) a(n)=12*n^2-8*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y For n > 0: odd terms in A001859.
%Y Cf. A001082.
%Y Cf. A016754, A000384.
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Dec 20 2012